Question: a) \(\log _{5} 3\), \(\log _{3} 5\)

Solution

The problem seems to involve evaluating or simplifying logarithmic expressions: Given expressions: \[ \log_5 3, \quad \log_3 5 \] Let’s start by expressing one logarithm in terms of the other using the change of base formula. The change of base formula is: \[ \log_a b = \frac{\log_c b}{\log_c a} \] Applying this formula to both logarithms: 1. Express \(\log_5 3\) using base 10: \[ \log_5 3 = \frac{\log_{10} 3}{\log_{10} 5} \] 2. Express \(\log_3 5\) using base 10: \[ \log_3 5 = \frac{\log_{10} 5}{\log_{10} 3} \] These are the expressions for \(\log_5 3\) and \(\log_3 5\) using a common logarithm (base 10). This transformation can be useful if you need to calculate or compare these values using a calculator that supports base 10 logs.